Gowers' Ramsey theorem for generalized tetris operations

Abstract

We prove a generalization of Gowers' theorem for ${\mathrm{FIN}}_k$ where, instead of the single tetris operation $T:{\mathrm{FIN}}_k\to {\mathrm{FIN}}_{k-1}$, one considers all maps from ${\mathrm{FIN}}_k$ to ${\mathrm{FIN}}_j$ for $0\le j\le k$ arising from nondecreasing surjections $f:\left\{0,1,\dots ,k+1\right\}\to \left\{0,1,\dots ,j+1\right\}$. This answers a question of Bartošová and Kwiatkowska. We also prove a common generalization of such a result and the Galvin--Glazer--Hindman theorem on finite products, in the setting of layered partial semigroups introduced by Farah, Hindman, and McLeod.

arXiv comment8 pages